The cart is given an initial push by a spring, and it comes to rest due to frict

Question

The cart is given an initial push by a spring, and it comes to rest due to friction.

Suppose the mass of the cart is mc = 304 g. You turn on the computer after the spring has been released, and stop it just as the cart comes to rest. The computer tells you that the equation of motion along the horizontal surface is:

x = 0.459 + 2.95t - 0.439t2

where x is the distance (in meters) the cart travels along the surface, and t is the time (in seconds). Notice that the numbers may not be realistic. Find:

  i. the magnitude of the acceleration of the cart: a =  m/s2

ii. k, the coefficient of kinetic friction between the cart and the surface: k =  

iii. tf, the time at which the cart comes to rest: tf =  s
HINT: Using the equation given, you should be able to write the equation of velocity vs. time.

Explanation / Answer

a)

Here , x = 0.459 + 2.95 t - 0.439 t^2

as v = dx/dt

v = d/dt(0.459 + 2.95 t - 0.439 t^2)

v = 2.95 - 0.878 * t

a = dv/dt

a = d/dt( 2.95 - 0.878 * t )

a = -0.878 m/s^2

magnitude of cart's acceleration is 0.878 m/s^2

ii) as a = uk * g

0.878 = 9.8 * uk

uk = 0.0896

the coefficient of kinetic friction between the cart and the surface is 0.0896

iii)

for cart to come at rest

v = 0

2.95 - 0.878 * t = 0

t = 3.36 s

the time at which cart is at rest is 3.36 s

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